DUKSUP: A Computer Program for High Thrust Launch Vehicle Trajectory Design & Optimization Spurlock, O.F.I and Williams, C. H.II NASA Glenn Research Center, Cleveland, OH, 44135 From the late 1960’s through 1997, the leadership of NASA’s Intermediate and Large class unmanned expendable launch vehicle projects resided at the NASA Lewis (now Glenn) Research Center (LeRC). One of LeRC’s primary responsibilities --- trajectory design and performance analysis --- was accomplished by an internally-developed analytic three dimensional computer program called DUKSUP. Because of its Calculus of Variations-based optimization routine, this code was generally more capable of finding optimal solutions than its contemporaries. A derivation of optimal control using the Calculus of Variations is summarized including transversality, intermediate, and final conditions. The two point boundary value problem is explained. A brief summary of the code’s operation is provided, including iteration via the Newton-Raphson scheme and integration of variational and motion equations via a 4th order Runge-Kutta scheme. Main subroutines are discussed. The history of the LeRC trajectory design efforts in the early 1960’s is explained within the context of supporting the Centaur upper stage program. How the code was constructed based on the operation of the Atlas/Centaur launch vehicle, the limits of the computers of that era, the limits of the computer programming languages, and the missions it supported are discussed. The vehicles DUKSUP supported (Atlas/Centaur, Titan/Centaur, and Shuttle/Centaur) are briefly described. The types of missions, including Earth orbital and interplanetary, are described. The roles of flight constraints and their impact on launch operations are detailed (such as jettisoning hardware on heating, Range Safety, ground station tracking, and elliptical parking orbits). The computer main frames on which the code was hosted are described. The applications of the code are detailed, including independent check of contractor analysis, benchmarking, leading edge analysis, and vehicle performance improvement assessments. Several of DUKSUP’s many major impacts on launches are discussed including Intelsat, Voyager, Pioneer Venus, HEAO, Galileo, and Cassini. Nomenclature C = first integral of Euler-Lagrange equations E = energy per unit mass e = eccentricity F = functional defined by Equation (3) f = thrust direction G = spherical Earth gravity constant g = gravity acceleration h = angular momentum per unit mass J = functional to be minimized m = mass N = total number of stages r = radius S = variational switching function T = thrust I Dep. Division Chief (retired), NASA Lewis Research Center, Advanced Space Analysis Office, 21000 Brookpark Rd., Cleveland, OH, 44135. The lead author is very recently deceased. II Aerospace Engineer, NASA Glenn Research Center, Propulsion Division, MS 142-5, 21000 Brookpark Rd., Cleveland, OH, 44135, Associate Fellow AIAA. 1 American Institute of Aeronautics and Astronautics t = time ν = velocity x = state variable z = vector pointing at north pole β = mass flow rate ε = jump factor η = fourth Lagrange multiplier λ = first Lagrange multiplier μ = second Lagrange multiplier σ = third Lagrange multiplier superscripts 0 = initial f = final ˙ = time derivative ˆ = unit vector subscripts 0 = initial f = final i,j,k,l,m,n = stage numbers p = perigee 1, 2 = components of oblate gravity acceleration I. Introduction his paper is a history of the rationale for, the development of, and the use of, a trajectory analysis code called TDUKSUP. It should be noted that it was very much a product of its time – over fifty years ago. While there is limited utility in describing it in detail since at this point it is of historical interest largely to students of trajectory analysis and optimization. Thus, this paper is not a “Users Guide”. There is value, however, in documenting how such a product of largely one individual had such a profound and sustained impact on so many of this nation’s launches of spacecraft. Though largely unknown to those outside of the launch vehicle trajectory design and performance optimization community, it was relied upon by NASA management to design the trajectories for most of the Atlas/Centaur, Titan/Centaur, and (anticipated) Shuttle/Centaur missions from the late sixties until 1997 with the launch of the Cassini mission to Saturn. Originally written in FORTRAN IV, the only technical computer language available at that time, DUKSUP retained that language’s coding through its lifetime. Because of its perceived value to the history of the NASA Lewis Research Center (LeRC) (now Glenn (GRC)), management included a hardcopy FORTRAN listing of the source code in the time capsule in front of the NASA Glenn Administration Building (#3); slated for opening in the year 2041, fifty years after its sealing. II. Background The Atlas/Centaur rocket was transferred to LeRC (later Glenn) Research Center (LeRC) in Cleveland, Ohio in the fall of 1962 following the failure of its maiden flight (designated “F-1”) managed by the Marshall Space Flight Center in Huntsville, Alabama. This took place within the backdrop of President Kennedy’s 1961 commitment to the United States landing a man on the moon by the end of the decade. Werner von Braun and the Marshall Space Flight Center were focused largely on responding to the challenge of building the launch vehicle to carry the crew to the Moon.1 The responsibility for solving Atlas/Centaur problems was then transferred to LeRC, due in large part to the technical expertise and determination of Dr. Abe Silverstein. Centaur became known as “Abe’s Baby”. The sole mission of the Atlas/Centaur at that time was to get a soft lander (Surveyor) to the moon prior to the attempt to land humans on the moon. A project office was established at LeRC to manage the development of the vehicle. Within 14 months of the transfer of the Centaur program to NASA LeRC, the problems were rectified and the first successful launch of an Atlas/Centaur launch vehicle (LeRC’s first attempt to launch an Atlas/Centaur), designated “AC-2”, occurred on November 27, 1963 (Fig. 1). The Lewis Research Center was the Power, Propulsion, and Communication technology research center for NASA. LeRC did not have experience with large projects such as this and had to find the staff mostly from research groups to meet the challenge. In anticipation that the Russians would be first to land humans on the moon due to the 2 American Institute of Aeronautics and Astronautics perceived lead that they had in the “race,” LeRC was doing the research on the propulsion for the vehicles to get to Mars. In 1961, a branch was created and staffed to provide the analytic resources to “guide” the research. Among the technologies being developed were a large hydrogen/oxygen engine, a large solid rocket motor, and a nuclear thermal rocket. Electric propulsion had been invented at LeRC and that research also continued. The analytic branch consisted of several sections that included thermal, structural, nuclear, and performance analysis capability (later control analysis was added). This analytic capability was in place or under development when the responsibility for the Atlas/Centaur was transferred to LeRC. As stated earlier, the research divisions were “raided” to staff the new project office. Significant parts of the analytic capability were transferred to the project office to support the challenging schedule of landing Surveyor on the moon prior to the human landing attempt. At that point, the performance analysis capability was not as mature as the other disciplines and was left in the Propulsion Research Division until sometime later. Initially, performance analysis for the Atlas/Centaur was done by the contractor, General Dynamics. For all practical purposes, LeRC had no capability to do vehicle performance analysis and it Figure 1. Launch of Atlas/Centaur-2 had to be developed from codes developed for other purposes, such as low thrust trajectory analysis for electric propulsion missions to Mars and an N-Body code available at LeRC.2 We IIIscrambled to develop vehicle performance capability at LeRC. The analysis for the Surveyor mission was done virtually entirely by the contractor using capability developed to support the Atlas ICBM program. They had analyzed the Surveyor mission with the Centaur for several years. By the late 1960’s, LeRC had developed the most sophisticated mission analysis and mission design capability in NASA and perhaps in the US. Obviously, we did not have access to what the classified world’s capabilities were. We were able to model the Centaur vehicles with such precision that post flight analysis verified performance capability to such accuracy that we could launch with minimal reserves. The trajectory designs were such that we could maximize the payload weights available to the spacecraft to the desired orbits. This was particularly true for geostationary orbit (GEO) missions where payload translated to profit or useful time on station. For the missions managed by LeRC, there evolved a philosophy and capability committed to extracting all the potential payload mass available from flying these optimum trajectories. In addition to optimizing the trajectories, constraints, such as instantaneous heating, were explicitly incorporated into the optimization. Applying constraints optimally can lower technical risk while minimizing the impact on payload capability. Fundamentally, this paper documents the history of how the NASA Lewis Research Center’s launch vehicle projects designed and flew the trajectories such that all the payload capability available from the vehicle was made available to the mission. III. Derivation of Optimum Control: the Calculus of Variations The following section outlines general mathematical theory on which DUKSUP was based. The Calculus of Variations (CoV) derivation as applied to this problem was previously published by the lead author.3 The appendix to that paper is included here for completeness. That derivation was the basis for the code. As new missions were added to the Atlas/Centaur manifest, the CoV analysis was extended to incorporate new requirements.